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| Formato: | Recurso digital |
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Zenodo
2026
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| Acceso en línea: | https://doi.org/10.5281/zenodo.18762423 |
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- <p><br><br>Why are there exactly three generations of matter? <br><br>We show the answer follows from one symmetry class. <br><br>Given a Hermitian circulant mass matrix with discrete ℤ/3ℤ family symmetry, we prove three results. <br><br>Theorem 1: the Koide formula Q = 2/3 is equivalent to a balance condition on the off-diagonal coupling. <br><br>Theorem 2: N = 3 is the unique dimension where this balance produces physical (all-positive) masses at the midpoint of the allowed range. <br><br>Theorem 3: the balanced spectrum depends on one scale and one phase angle. <br><br>The companion paper (<em>Selecting the Koide Phase</em>) derives that angle from the self-referential axiom σ = 1/(1+σ), yielding cos(δ) = −19/28 with zero fitted parameters and reproducing all three charged lepton masses to 0.021% total error. <br><br>Part of the Pentagon Physics programme.</p>